Ordinary Differential Equations with Applications to Mechanics

496 Pages·2007·6.67 MB·English

by  Mircea V. Soare, Petre P. Teodorescu, Ileana Toma (auth.)

Ordinary Differential Equations
with Applications to Mechanics
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume585
Ordinary Differential
Equations with
Applications to Mechanics
by
Mircea V. Soare
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Petre P. Teodorescu
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and
Ileana Toma
(cid:84)(cid:101)(cid:99)(cid:104)(cid:110)(cid:105)(cid:99)(cid:97)(cid:108)(cid:32)(cid:85)(cid:110)(cid:105)(cid:118)(cid:101)(cid:114)(cid:115)(cid:105)(cid:116)(cid:121)(cid:32)(cid:111)(cid:102)(cid:32)(cid:67)(cid:105)(cid:118)(cid:105)(cid:108)(cid:32)(cid:69)(cid:110)(cid:103)(cid:105)(cid:110)(cid:101)(cid:101)(cid:114)(cid:105)(cid:110)(cid:103)(cid:44)
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AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-5439-4 (HB)
ISBN-13 978-1-4020-5439-6 (HB)
ISBN-10 1-4020-5440-8 (e-book)
ISBN-13 978-1-4020-5440-2 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
Translated into English, revised and extended by Petre P. Teodorescu and Ileana Toma
All Rights Reserved
© 2000 EDITURA TEHNICĂ
This translation of “Ordinary Differential Equations with Applications to Mechanics”
(original title: Ecuatü, diferentiale cu aplicatii îa mecanica constuctübor, published by:
EDITURA TEHNICĂ, Bucharest, Romania, 1999), First Edition, is published by
arrangement with EDITURA TEHNICĂ, Bucharest, Romania
© 2007 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
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CONTENTS

PREFACE ix
INTRODUCTION 1
1. Generalities 1
2. Ordinary Differential Equations 3
3. Supplementary Conditions Associated to ODEs 5
3.1 The Cauchy (initial) problem 5
3.2 The two-point problem 9
1. LINEAR ODEs OF FIRST AND SECOND ORDER 11
1. Linear First Order ODEs 11
1.1 Equations of the form y′= f(x) 11
1.2 The linear homogeneous equation 12
1.3 The general case 12
1.4 The method of variation of parameters (Lagrange’s method) 13
1.5 Differential polynomials 15
2. Linear Second Order ODEs 16
2.1 Homogeneous equations 17
2.2 Non-homogeneous equations. Lagrange’s method 20
2.3 ODEs with constant coefficients 24
2.4 Order reduction 27
2.5 The Cauchy problem. Analytical methods to obtain the solution 29
2.6 Two-point problems (Picard) 31
2.7 Sturm-Liouville problems 33
2.8 Linear ODEs of special form 36
3. Applications 43
2. LINEAR ODEs OF HIGHER ORDER (n>2) 131
1. The General Study of Linear ODEs of order n>2 131
1.1 Generalities 131
1.2 Linear homogeneous ODEs 131
1.3 The general solution of the non-homogeneous ODE 136
1.4 Order reduction 136
2. Linear ODEs with Constant Coefficients 137
2.1 The general solution of the homogeneous equation 138
2.2 The non-homogeneous ODE 141
2.3 Euler type ODEs 143
3. Fundamental Solution. Green Function 143
3.1 The fundamental solution 143
3.2 The Green function 144
v
vi ODEs WITH APPLICATIONS TO MECANICS

3.3 The non-homogeneous problem 146
3.4 The homogeneous two-point problem. Eigenvalues 147
4. Applications 148
3. LINEAR ODSs OF FIRST ORDER 209
1. The General Study of Linear First Order ODSs 209
1.1 Generalities 209
1.2 The general solution of the homogeneous ODS 210
1.3 The general solution of the non-homogeneous ODS 211
1.4 Order reduction of homogeneous ODSs 212
1.5 Boundary value problems for ODSs 213
2. ODSs with Constant Coefficients 215
2.1 The general solution of the homogeneous ODS 215
2.2 Solutions in matrix form for linear ODSs with constant coefficients 217
3. Applications 221
4. NON-LINEAR ODEs OF FIRST AND SECOND ORDER 239
1. First Order Non-Linear ODEs 239
1.1. Forms of first order ODEs and of their solutions 239
1.1.1 Forms of ODEs 239
1.1.2 Forms of the solutions 239
1.2 Geometric interpretation. The theorem of existence and uniqueness 241
1.3 Analytic methods for solving first order non-linear ODEs 245
1.4 First order ODEs integrable by quadratures 247
1.4.1 ODEs with separate variables 247
1.4.2 ODEs with separable variables 248
1.4.3 Homogeneous first order ODEs 248
1.4.4 ODEs of the form 249
1.4.5 Total differential ODEs 249
1.4.6 Integrant factor 251
1.4.7 Clairaut’s equation 254
1.4.8 Lagrange’s equation 255
1.4.9 Bernoulli’s equation 256
1.4.10 Riccati’s equation 257
2. Non-linear Second Order ODEs 260
2.1 Cauchy problems 260
2.2 Two-point problems 260
2.3 Order reduction of second order ODEs 261
2.4 The Bernoulli-Euler equation 263
2.5 Elliptic integrals 265
3. Applications 268
Contents vii
5. NON-LINEAR ODSs OF FIRST ORDER 365
1. Generalities 365
1.1 The general form of a first order ODS 365
1.2 The existence and uniqueness theorem for the solution of the
Cauchy problem 366
1.3 The particle dynamics 367
2 . First Integrals of an ODS 369
2.1 Generalities 369
2.2 The theorem of conservation of the kinetic energy 371
2.3 The symmetric form of an ODS. Integral combinations 372
2.4 Jacobi’s multiplier. The method of the last multiplier 373
3. Analytical Methods of Solving the Cauchy Problem for Non-Linear ODS s 376
3.1 The method of successive approximations (Picard-Lindelõff) 376
3.2 The method of the Taylor series expansion 376
3.3 The linear equivalence method (LEM) 378
3.3.1 Solutions of non-linear ODSs by LEM 381
3.3.2 New LEM representations in the case of polynomial coefficients 382
4. Applications 383
6. VARIATIONAL CALCULUS 415
1. Necessary Condition of Extremum for Functionals of Integral Type 415
1.1 Generalities 415
x2
1.2 Functionals of the form I[y]≡ ∫F(x,y(x),y′(x))dx 417
x1
1.3 Functionals of the form I[y]≡ x∫2F(x,y,y′,y′′,...,y(n))dx 418
x1
1.4 Functionals of integral type, depending on n functions 419
2. Conditional Extrema 421
2.1 Isoperimetric problems 421
2.2 Lagrange’s problem 423
3. Applications 426
7. STABILITY 451
1. Lyapunov Stability 451
1.1 Generalities 451
1.2 Lyapunov’s theorem of stability 452
viii ODEs WITH APPLICATIONS TO MECANICS
2. The Stability of the Solutions of Dynamical Systems 454
2.1 Autonomous dynamical systems 454
2.2 Long term behaviour of the solutions 456
3. Applications 458
PROBLEM INDEX 483
REFERENCES 485
PREFACE
The present book has its source in the authors’ wish to create a bridge between
mathematics and the technical disciplines that need a good knowledge of a strong
mathematical tool. The authors tried to reflect a common experience of the University of
Bucharest, Faculty of Mathematics and of the Technical University of Civil Engineering
of Bucharest.
The necessity of such an interdisciplinary work drove the authors to publish a first book
with this aim (“Ecuaţii diferenţiale cu aplicaţii în mecanica construcţiilor” – Ordinary
differential equations with applications to the mechanics of constructions, Editura
Tehnică, Bucharest, Romania).
The present book is a new edition of the volume published in 1999. Unfortunately, the
first author (M.V. Soare) passed away shortly before the publication of the Romanian
edition, so that the present work is only due to the other two authors. It contains many
improvements concerning the theoretical (mathematical) information, as well as new
topics, using enlarged and updated references.
We considered only ordinary differential equations and their solutions in an analytical
frame, leaving aside their numerical approach.
Compared to the Romanian edition, this volume presents the applications in a new way.
The problem is firstly stated in its mechanical frame. Then the mathematical model is set
up, emphasizing on the one hand the physical magnitude playing the part of the unknown
function and on the other hand the laws of mechanics that lead to an ordinary differential
equation or system. The solution is then obtained by specifying the mathematical
methods described in the corresponding theoretical presentation. Finally – last, but not
least – a mechanical interpretation of the solution is provided, this giving rise to a
complete knowledge of the studied phenomenon; after all, this is the main goal of any
scientific approach. In most of cases, the solution is interpreted by using a parametrical
study, which better emphasizes the core of the phenomenon. Sometimes, we pointed out
the influence of a certain parameter or presented auxiliary diagrams and tables, whence,
by interpolation, one can immediately get effective numerical values of the solution.
The number of the applications was increased; in order to keep the volume within a
reasonable number of pages and also, not to exaggerate the interference between
mathematics and engineering, we did not exhaustively introduce and present the
mathematical model. It must be pointed out that many of these problems currently
appear in engineering.
ix
x ODEs WITH APPLICATIONS TO MECHANICS
The book is organized in seven chapters. Each of them begins with a theoretical
presentation, which insists on the practical computation – the “know-how” of the
mathematical method – and ends with a rich range of applications. Unlike the standard
presentations, we introduced separately the linear case, which is exposed in the first three
chapters. The reason of this is that in the linear case one can use not only general
methods, fitted for any differential equation, but also specific methods. The non-linear
case forms the object of the next two chapters. The sixth chapter treats problems in a
variational frame. Finally, the last chapter is devoted to an initiation in the modern
domain of stability.
It should be mentioned that the book contains some personal results of the authors,
published in scientific reviews of wide circulation.
The prerequisites of this book are courses of elementary analysis and algebra, acquired
by a student in a technical university. It is addressed to a large audience, to all those
interested in using mathematical models and methods in various fields, like: mechanics,
civil and mechanical engineering, people involved in teaching or design as well as
students.

P.P.TEODORESCU and ILEANA TOMA